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Modeling of High-Current Arcs with Emphasis on Free Surface Phenomena in the Weld Pool Calculations suggest that flow patterns in deformed weld pools may be more complex than previously thought BY R. T. C. C H O O , J. SZEKELY A N D R. C. WESTHOFF ABSTRACT. An extensive set of calculations is presented to describe the effect of the role played by the arc weld pool interface in affecting both the arc and the weld pool behavior. The computed results may be classified into the following three main groups: 1) The behavior of the welding arc for both flat and deformed weld pool surfaces. Here, the computed results were found to be in excellent agreement with measurements reported for flat weld pool surfaces or flat anodes in general. However, it was shown that when the free surface of the weld pool is significantly deformed, the resultant heat and current flux falling on the anode may be markedly modified. It follows that the usually postulated Gaussian heat flux distribution may not be generally applicable for significantly deformed weld pools. 2) Computed results reported on heat flow and melt circulation involving deformed weld pools have shown that under these conditions the melt circulation may be markedly affected by the shape of the free surface. Indeed, it has been found that deformed free surfaces may result in very complex weld pool circulation patterns. In the present case, the actual free surface shape was deduced from previous experimental observations. Since the heat flux and current distribution employed at the weld pool surface were obtained from the previously described arc calculations, the linkage between arc and weld pool behavior has been accomplished. A further refinement, which is being currently pursued, would allow the calculation of the deformed weld pool surface. 3) Finally, calculations have been presented describing the transient collapse of R. T. C. CHOO, j SZEKELY and R. C WESTHOFF are with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 346-s I SEPTEMBER 1990 a weld pool assuming isothermal conditions as an approximate representation. These preliminary results are able to capture some essential features of experimentally observed weld pool collapse, including the entrapment of gases and the expulsion of liquid droplets. Introduction The role of convection in arc welding operations and its effects on the structure and properties of the resultant weld is well documented (Refs. 1-4). Convection in weld pools is driven by a combination of forces, which include surface tension gradients, buoyancy, electromagnetic forces and the shear stress exerted by the gas on the molten pool surface. Indirectly, all these forces are derived from the arc, which is struck between the electrode and the workpiece —Fig. 1. The spatial distribution of the energy flux falling on the weld pool surface will have a marked effect on the weld pool shape and on the subsequent solidification process, which in turn affects the structure and properties of the weldment produced. It follows that the interface between the welding arc and the free surface of the weld pool represents a critical component of the welding operation. Up to the present time, most investiga- KEY W O R D S High-Current Arcs Free Surface Phenomenon Weld Pool Arc Weld Pool Interface Arc Behavior Weld Pool Behavior Deformed Weld Pool Weld Pool Collapse Modeling Welding Arc Modeling tors have concentrated on representing weld pool behavior on the one hand and the modeling of welding arcs on the other, with relatively little attention being paid to the interfacial regions. Indeed, in virtually all previously published papers concerning weld pool behavior, this interfacial region was introduced in terms of the boundary conditions, with a Gaussiantype current and heat flux distribution being specified. By the same token, in previous studies of welding arc phenomena, the weld pool was represented as a flat surface having a constant temperature and electric potential. There is persuasive qualitative evidence that these "standard" postulates may represent a gross oversimplification in many instances. Thus, in the case of deformed weld pools, which are observed for operation at high current levels, there may be an important two-way interaction between the welding arc and the molten regions in that the nature of the arc may be affected by the pool shape and viceversa; and so heat transfer to the pool may be modified by changes in the arc. The marked nonlinearity and hysteresis behavior in the relationship between surface depression and arc current (Fig. 2) reported by Lin and Eagar (Ref. 5) provide clear support for this contention. Yet other, more complex forms of interaction may involve ripple or wave formation at the free weld pool surface, the existence of which has been reported, but not analyzed. It is suggested that a comprehensive representation of these free surface phenomena could play an important role in providing greatly improved insight into weld pool behavior and could represent an interesting new frontier in the modeling of arc welding systems. At the same time, these phenomena could be relevant also to a range of other joining processes, including electron beam and laser beam welding operations. To provide a realistic perspective, let us Electrode (Cathode) 5 - 1 - c o Plasma Arc Base Metal (Anode) Molten Pool 240 260 280 300 Current (A) Fig. 2— Variation of surface depression depth with current. The arrow indicates the direction in which the current was increased and decreased (Ref. 5). Fig. 1 — Schematic of arc welding operation. briefly review recent relevant work concerned with the modeling of welding operations. In the last five years, there have been major advances made in the numerical simulation of the weld pool (Refs. 6 11). Equally important, parallel developments regarding arc behavior have allowed the numerical representations of current density, temperature and velocity in the welding arc (Refs. 12, 13). At the present time, the representation of fully three-dimensional weld pool circulation has become an accepted fact (Refs. 14, 15); furthermore, a very elegant comprehensive modeling of continuous welding operations in three dimensions has also been accomplished (Refs. 16, 17). As a result of these works, predictions may now be made with confidence regarding the transient temperature and velocity profiles in weld pools, provided the boundary conditions may be defined at the weld pool-plasma arc interface. As a practical matter, all these models are being run by specifying the free surface shape and the current and heat flux distributions at the free surface. Notable exceptions are the recent work of Zacharia, ef al. (Refs. 16, 17), and that of Paul and DebRoy (for laser beam welding) (Ref. 18), which allow free surface deformation, but again, for a set of thermal boundary conditions that are specified a priori. The main purpose of the present article is to address a relatively unexplored but potentially important problem area; that of the free surface behavior of welding arc and by implication, free surface problems in welding operations in general. The ultimate objective of the work is to develop an understanding and quantitative representation of the dynamic, twoway coupling between the transport phenomena in the welding arc and in the weld pool. The results to be presented here represent an intermediate stage in this direction but a further development of ideas first suggested in 1986 (Ref. 19). More specifically, we shall calculate arc parameters for certain specified weld pool shapes taken from experimental data, and then use the computed heat and current fluxes falling on the free weld pool surface to evaluate the temperature and the velocity profiles in the weld pool. The important consequence of this approach is that allowance can be made for nonGaussian heat and current fluxes falling on the free surface of the weld pool. Free Surface Problems in Welding The schematic sketch given in Fig. 1 was an oversimplification because the upper surface of the weld pool was represented as an undistorted plane. While the postulate of a flat weld pool surface may be appropriate for certain types of operations, it is known that significant weld pool deformation may take place in arc welding at high current levels, in submerged arc welding, in laser and EB welding and in Oscillating Weldpool U) GMA operations. The free surface deformation of weld pools may have several important practical consequences, as sketched in Fig. 3 (Refs. 20, 21): 1) At significant deformation levels the welding arc itself may be affected by the weld pool shape, and thus the heat and current flux may be markedly modified by the weld pool behavior. 2) The interaction of the plasma gas stream with the weld pool may cause surface ripples, oscillating surfaces or instabilities. 3) The collapse of a significantly deformed weld pool may produce numerous weld defects including keyholes, gas bubble entrapment and the like. The issues addressed in this paper are the modeling of the welding arc with a predeformed anode surface, the modeling of the fluid flow and heat transfer in the depressed weld pool, and the modeling of the collapse of a keyhole depression in the weld pool. The effect of instabilities will be discussed in a separate publication (Ref. 22). R i p p l i n g Effects D e p r e s s e d Surface (b) Fig. 3 — Effects of free surface on weld pool. A —CTAW can produce oscillating weld pool surface (Ref. 20); B — rippling effects can be seen in moving welds (Ref. 21); C — free surface is depressed at high arc currents (Ref. 5). WELDING RESEARCH SUPPLEMENT 1347-s Modeling of the Welding Arc with a D e f o r m e d Anode Surface Figure 4 shows the computational domain used to model the arc. Excellent reviews of welding arc behavior are available in the literature (Refs. 23, 24). In the present case, w e shall briefly summarize the key features of arc behavior. When an electric potential difference is set up between the anode (metal workpiece) and the cathode (usually a tungsten rod), and a spark or some initial heating ionizes the gas between the two, an arc may form. The ionized gas, or plasma, has a much higher electrical conductivity than the gas in a normal state, which allows current to flow. This flow of current causes joule heating in the gas, allowing it to maintain its ionized state so that it remains electrically conductive; and the arc becomes self-sustaining. As the electrical conductivity of the plasma is highly temperature dependent, the arc is constrained to a narrow path between the cathode and anode, and the current density within the arc column may assume a Gaussianshaped radial distribution. At the cathode, electrons are thermally emitted from a small, hot region known as the cathode spot. From the cathode spot the electrons diverge radially and move axially toward the anode. The strong divergence of the current at the cathode gives rise to significant electromagnetic or Lorentz forces which drive the cathode jet, a high-speed flow of gas, toward the anode. The combination of the cathode jet and joule heating contribute to the convective heat flux and pressure at the anode surface. At the anode, the electrons are condensed, giving up energy proportional to the work function of the anode material. This electron component of the anode heat flux is usually about 70% of the total anode heat flux. The following assumptions will be made in this analysis: 1) The arc is assumed to be radially symmetrical. 2) Steady-state conditions are assumed throughout the arc. 3) The arc is assumed to be in local thermodynamic equilibrium (LTE), which is taken to mean that the electron and heavy particle temperatures are not significantly different. The studies by Hsu and Pfender (Refs. 12, 25) show this assumption is accurate through most of the arc, except in the fringes and very near the cathode and anode surfaces. 4) The arc plasma is assumed to consist of pure argon at atmospheric pressure. The effect of other gases that may be entrained is neglected as is the effect of metal vapors from the electrode and workpiece. 5) The flow is assumed to be laminar. This assumption was justified by McKelliget and Szekely (Ref. 13) on the basis of 348-s I SEPTEMBER 1990 laminar-turbulent transition for a free jet. 6) The plasma is assumed to be optically thin so that radiation may be accounted for using an optically thin radiation loss per unit volume. 7) The heating effects of viscous dissipation, and buoyancy forces due to gravity are neglected. Governing Transport Equations Using the above assumptions, the governing equations for the arc may be written as follows: Conservation of mass: 7 Jr (prUr) + Tz ^,) =0 d) Conservation of radial momentum: 2 r iKf)]-' 2u r (2) az Conservation of axial momentum: ±|(pruruz) + JjP 'dz oz (PUZ 2 ) = + 1_5 rdr f)]+ dz 2 -dzf (?) + )rl Conservation of thermal energy: 1 1 (pru h) r r or + (puzh) = /d<_ db\ d_ ( k d h \ r or \ C p d r j + dz \ C p d z / p 5 kb SR 2 e 1 A I J_ dh 5h C p 5z + )r Cp fir + (4) The source terms in braces in the energy equation represent joule heating, radiation losses and transport of enthalpy due to electron drift, respectively. Current continuity in terms of electric potential: 1 d ( dA.df dA 7 a? ^ r a F J + az {° Tz) = For the purpose of modeling, the welding arc shown in Fig. 1 is transformed to the domain shown in Fig. 4 and the corresponding boundary conditions are given in Table 1. These specify zero velocities at solid boundaries, zero fluxes at the axis of symmetry, constant temperature of 1000 K at inflow boundaries, a zero radial temperature gradient at the outflow boundary, zero currents at all boundaries except at the cathode spot and at the anode, where the current density and the potential are specified, respectively; a constant electric potential is specified at the anode. Most of these boundary conditions are self-evident but a few require some discussion. Hsu, ef al. (Ref. 12), investigated the effect of the temperature boundary conditions at the inflow region DE and reported that the arc characteristics remained substantially unchanged whether a 1000 K or a 2000 K isotherm was used. This observation is probably due to the fact that the specific heat of the argon gas in this temperature range does not vary significantly. Secondly, the choice of zero radial temperature gradient at the outflow region FG assumes that this region is far away from the plasma column such that the radial temperature gradient approaches zero. The anode and cathode surfaces require special treatment since deviations from LTE occur in these regions and these are given below. Cathode Region 0{5) Since the current distribution is axisymmetrical, the self-induced magnetic field is given by the following relation from Ampere's law: =7 >dr (7) Boundary Conditions dP (pUrUz) = - -Jjj + h^+k -oV</> The physical properties, namely density, viscosity, thermal conductivity, heat capacity and electrical conductivity, are treated as temperature dependent in the model. Compressibility effects have been neglected, which is admissible if the plasma velocity is significantly lower than the speed of sound. This assumption was tested by including the compressibility terms of the momentum equations, and little or no effect was noticed in the flow field. The temperature-dependent plasma properties were taken from the tabulated data of Liu (Ref. 26), and the radiation loss is adapted from the measurements of Evans and Tankin (Ref. 27). (6) Current density is calculated from the definition of electric potential: Between the arc column and the cathode surface there is a thin transition layer in which steep gradients occur; it supports several of the physical processes that sustain the arc. Positive ions are accelerated toward the cathode surface and provide energy for the thermal emission of electrons. The emitted electrons either combine with the positive ions or are accelerated away from the cathode. Additionally, the plasma is ionized in this layer, causing a substantial potential drop called the Inflow cathode fall. This cathode boundary layer has been investigated by Hsu and Pfender (Ref. 25) who showed that as the cathode surface is approached, the heavy particle temperature approaches the cathode surface temperature while the electron temperature remains much higher (~17,000 K). This state of thermal nonequilibrium cannot be represented using an LTE model. In addition, the thickness of the cathode boundary layer is of the order 0.1 mm, which presents some practical problems in being resolved using a 2-D finite-difference approximation. The contribution of the cathode fall has been approximately accounted for using a free-fall type of expression for the cathode fall voltage, V c , as was done by McKelliget and Szekely (Ref. 13): . , _ 5 kbTe|ec V c ~2 e ' Qioniz = | J c | V c (8) Here, Qbniz is a positive source to the plasma column at the cathode boundary which approximates the energy used in the cathode boundary layer to ionize the plasma. Te|ec is approximated as the maximum plasma temperature in the column adjacent to the cathode ( ~ 20,000 K), so V c ~ 4.3 V. The boundary condition for the electric potential is approximated assuming that the cathode current density, Jc, emitted from the cathode normal to the surface is constant inside the cathode spot radius, Rc, and is zero outside: (9) 0 Inflow - ^ r>Rc The unknown parameter Jc must be specified by giving the cathode spot radius. McKelliget and Szekely (Ref. 13) showed that a single value of Jc (6.5 X 107 A/m 2 ) could be used over a range of conditions. Anode Region Four principal modes of heat transfer contribute to the anode heat flux in welding: convection from the plasma, electron flow due to the current, radiation from the plasma, and vaporization of the anode. The boundary layer which exists between the anode and the arc column is of special interest as the processes which occur there govern the current density and heat flux to the anode. Dinulescu and Pfender (Ref. 28) presented a model of the anode boundary layer which showed that the heavy particle temperature approaches the anode temperature near the wall, while the electron temperature remains high ( ~ 10,000 K), thus maintaining a conducting path to the anode. This boundary layer region is approximately 0.1-mm (0.004-in.) thick and presents the same problems to an LTE model as the cathode boundary layer. W e have thus chosen to take the same approach used Outflow Fig. 4 —Domain of integration for the welding arc. 10,000 K, the heat flux due to electron flow may be written as shown: by McKelliget and Szekely (Ref. 13). The anode fall is assumed to be negative as seen by the modeling (Ref. 28) and experiments (Ref. 29) of Pfender and coworkers so that the electron contribution to the anode heat flux may be written as follows: Kb I elec Q, + LV» Q e = | a (2.76 + V w ) (10) 4>a is the thermal diffusion coefficient of the electrons. The quantity in parentheses is assumed to be 3.203, corresponding to an argon plasma with electron temperatures of approximately 10,000 K. If the electron temperature is assumed to be (11) where V w is the work function of the anode material. The approach taken here is to assume that the electron temperature is equal to the film temperature, Tf = 0.5(TW + Te), where T e is the temperature at the edge of the boundary layer and T w is the temperature of the wall. This is an approximate way to allow for the fact that the electron temperature will be lower as the radius increases near the anode. This assumption gives an electron temperature Table 1 - Boundary Conditions for the Integration Domain Shown in Fig. 4 BC CD DE T = 3000 K 0 0 0 0 T = 3000 K 0 0 T = 3000 K 0 EF (inflow) dpru r FG (outflow) dprur CH h u2 Ur AB ~~d~r~ T = 1000 K az 0 0 ^ =0 ar ar 0 Q-IJcNc T = 1000 K ar 0 T = 1000 K Equation 8 <p ,c = ^7 8* - • az ar ar 0 = constant HA 0 ^ = 0 ar ar WELDING RESEARCH SUPPLEMENT | 349-S and w r i t t e n as f o l l o w s : = 0515 Pr w ( a. O Table 2—Variable Mesh Used for the Four Cases Studied for Welding Arc Modeling /MePeV" VMWPW/ S *Y n, IPwPw— I (he - u i (13) hw) _i UJ > x o The correlation as given in the literature is a p p a r e n t l y only valid at t h e stagnation point (i.e., at r = 0). It has b e e n m o d i f i e d f o r use o v e r the entire a n o d e b y replacing ce < t h e radial v e l o c i t y gradient —r— by —. This UJ a ui v> ce is d o n e because the radial velocity at the a n o d e goes t h r o u g h a m a x i m u m at s o m e . . , ,. dure critical radius, so that —r— goes t o z e r o 5 cu w h i l e t h e c o n v e c t i v e heat flux remains f i - _J UJ nite. T h e t e r m — is selected because it a p - Ul a proaches Ul o > x o o: < ui 05 Ul o: £ a. o -J Ui > Ul a Fig. 5 - Configuration of radiation view factors. o f a p p r o x i m a t e l y 7000 K f o r a 1000 K anode. The c o n v e c t i v e c o n t r i b u t i o n c o u l d b e f o u n d b y a p p l y i n g a correlation f o r a stagnation p o i n t f l o w . H e r e , a m o d i f i e d version o f the a p p r o a c h taken b y M c K e l liget and Szekely (Ref. 13) is used. The same c o n v e c t i o n correlation w a s taken f r o m the literature (Refs. 30, 31). Nuv = 0.515 PePe (12a) MwPw Ul t/i Nuw z \ (h e - hv Pr„ ui S a, O > ui as r —• 0 and as , t h e n Equation 12a can be m a n i p u - •»> x o a: < — j — both r —• oo, and assumes a reasonable f o r m i n - b e t w e e n . Additionally, if the Reynolds n u m b e r , R e w , is assumed t o b e g i v e n b y F~ V dure (12b) MwPw-^T T h e radiative heat flux t o the a n o d e is calculated using the f o l l o w i n g relationship—Fig. 5: '<> = J 477?cos ^ d V i (14) V| a <r: Ul! z ui £ a Fig. 6 — Temperature profile for a 200-A arc and 100-mm arc length. Predictions compared with experimental results of Pfender, et al. (Ref. 12). The a b o v e e q u a t i o n m a y b e w r i t t e n as an elliptic integral in the 9 d i r e c t i o n , the solution of w h i c h can b e f o u n d f r o m an integral table (Ref. 32). It is t h e n integrated numerically in the r a n d z directions, o v e r the entire calculation d o m a i n . T h e ass u m p t i o n e m p l o y e d h e r e is that the v o l u m e o f the cells used in the calculation is small e n o u g h t o a p p r o x i m a t e a differential volume. In this analysis, heat transfer d u e t o va- Radial Position ( m m ) > x _l ui > ui 0 - a o ce < UJ (/> Experiment Ul ce 3 5 0 - s | SEPTEMBER 1990 200 260 280 300 1.0 1.2 1.8 4.0 19 X 37 19 X 38 19 X 41 19X43 Domain Dimensions (mm) 10.95 10.95 10.95 10.95 X X X X 11.7 11.9 12.5 14.7 p o r i z a t i o n o f the a n o d e metal has b e e n n e g l e c t e d . This assumption is realistic if w a t e r - c o o l e d anodes are used. In actual w e l d i n g c o n d i t i o n s , v a p o r i z a t i o n of t h e a n o d e material c o u l d b e c o m e i m p o r t a n t and m a y significantly change b o t h t h e c o n d u c t i v i t y a n d the radiation losses o f the plasma near the a n o d e . A n o d e v a p o ration m a y b e r e p r e s e n t e d a p p r o x i m a t e l y using a mass-transfer analogy t o c o n v e c tive heat transfer. A n o d e v a p o r a t i o n is n o t c o n s i d e r e d in this analysis. The heat flux t o the a n o d e surface can t h e n b e w r i t t e n as, (15) T h e a n o d e is assumed t o b e a p e r f e c t c o n d u c t o r relative t o the plasma so that a constant value of <fr is used f o r the b o u n d ary c o n d i t i o n at the a n o d e surface. T h e g o v e r n i n g equations and b o u n d a r y conditions w e r e solved using a finite-volu m e a p p r o a c h d e s c r i b e d b y Pun a n d Spalding (Ref. 32) w h i c h w a s i m p l e m e n t e d using a m o d i f i e d version o f t h e 2/E/FIX c o d e . The difference equations w e r e solved by iteration until t h e y w e r e satisfied w i t h i n 99.9%. Several variable meshes w e r e used d e p e n d i n g o n the arc length d e p t h of the a n o d e depression. These are listed in Table 2. T h e 300-A case takes a b o u t t w o hours CPU t i m e o n a VaxStation II. Results and Discussions of the Welding Arc Modeling — o o Grids (radial x axial) Solution Technique Ci CJ Surface Depression (mm) Q a (r) = Q e (r) + Q c (r) + Q r (r) lated i n t o the f o r m o f Equation 13. Q Arc Current (A) Figures 6 a n d 7 s h o w a c o m p a r i s o n o f the theoretical predictions w i t h e x p e r i mental measurements of Hsu, et al. (Ref. 12), a n d N e s t o r (Ref. 34), f o r the t e m p e r ature profiles, and t h e c u r r e n t density a n d heat flux, respectively. T h e g o o d agreem e n t p r o v i d e s a calibration f o r t h e c o m putational a p p r o a c h . Figures 8 - 1 1 s h o w the current density flux, heat flux, t e m p e r a t u r e profile and velocity p r o f i l e , respectively, f o r w e l d pools w h i c h h a v e b e e n depressed by 1.0 m m (0.04 in.) (200 A), 1.2 m m (0.05 in.) (260 A), 1.8 m m (0.07 in.) (280 A), a n d 4.0 m m (0.16 in.) (300 A). Their c o r r e s p o n d i n g calculated arc parameters are s h o w n in Table 3. W e n o t e that the w e l d p o o l d e - 80 I _, , , , 1 , '— CO ta a 2 — : 200 AMP ABC L = 6.3 MM x — Calculation Voltage - 16.4 Efficiency = 0.59 Experiment (Nestor) \ \ \ ~ ; ^ i i -i—i—i—i—r — 200 AMP ARC L = 8.3 MM r - Total Flui * - Qrad + - Qelectron o — QconT Experiment (Nestor) — — • \\ i ^^^^ i i 1 i i i i rr^r^-rrH—i—*mt 2 i 4 6 RADIAL LOCATION (MM) 8 ~ " nir" «*-*• 10 2 10 8 4 6 RADIAL LOCATION (MM) Fig. 7— Current density and heat flux distribution at anode for a 200-A arc. Predictions compared with experimental results of Nestor (Ref. 34). pression itself was not calculated from the model, but reflects experimental measurements reported by Lin and Eagar (Fig. 2) (Ref. 5). In Figs. 8 and 9, a bimodal distribution (only one-half is seen because of symmetry) is observed for the heat and current fluxes. This is in sharp contrast with Figs. 6 and 7 in which a Gaussian distribution is observed for a flat anode. As the current increases, the bimodal distribution becomes more pronounced due to the deeper surface depression. The effect of this surface depression is quite dominant. For the 280-A case, we see that the peak current density is four times that at the center of the pool, while the 300-A case is almost ten times that of the center. Since about 70% of the thermal energy transferred to the anode is carried by the electrons, this change in the heat flux may have a marked effect on both shape of the melt-solid boundary and on convection within the weld pool. flux, but the ratio of the maxima to the minima is not as large because the deformed pool affects the convective contribution much less than it does the electronic contribution. As the current increases, the gas jet velocity also increases; so that we see a rather flat heat flux profile in the middle region of the weld pool (r < 1.5 mm) as convection continues to add some heat to the center of the pool. The radiative contribution to the total heat flux is generally small ( ~ 10%). The important finding represented by these results is The heat flux to the anode shows a bimodal distribution similar to the current n 55 W Q 2 4 6 RADIAL LOCATION (MM) 2 . " " I 4 6 RADIAL LOCATION (MM) 1 1 ! 1 | T 1 /-* f 3 — — - 1 I l I" 1 T • "I" T 1 4 m m DEPRESSION T ' "- — Voltage = 18.4 \ Efficiency = 0.66 1 T — r - Calculation \ 1 D; 300 AMP ARC \ / T Q — *T7. 2 4 6 RADIAL LOCATION (MM) ,1 , i 2 i i 1 i i i i 1 i 4 6 RADIAL LOCATION (MM) i i i 1 X-r 10 Fig. 8-Anode current density for the four cases studied. A-200 A; B-260 A; C-280 A; D-300 A. WELDING RESEARCH SUPPLEMENT | 351-s Table 3—Calculated Arc Parameters for the Four Cases Studied'3' 280 Arc current (A) 200 260 300 369 402 Max. velocity 298 421 (m/s) 21,800 23,100 23,600 23,: Max. temperature K Max. anode 934 1080 565 1140 pressure (Pa) Max. current 5.5 3.9 4.2 3.6 density (A/mm 2 ) Max. heat tlux 35.0 48.3 39.9 30.5 (W/mm2) Arc voltage 16.4 17.6 18.0 18.4 (V) 59 Arc efficiency 59 58 56 (%) (a) The arc length is defined al 6.3 m m which is the distance f r o m the tip of the electrode to the level surface of the w o r k piece. that the actual heat flux to the anode may markedly depend on the shape of the free weld pool surface. Although the total heat flux for the 280A case (Fig. 9C) appears larger than the 300-A case (Fig. 9D), this is due to the fact that the surface area of the depressed pool for the 300-A case (4.0 mm/0.16 in. depression) is much larger than the 280-A 2 case (1.8 mm/0.07 in. depression). In fact, the total heat flux received by the 300-A case is about 10% larger than the 280-A case. The temperature contours shown in Fig. 10 and the maximum temperatures given in Table 3 illustrate the nature of the temperature field in an arc adjacent to a depressed pool. The isotherms are very close near the cathode because the cathode must be maintained at 3000 K — near the melting temperature of tungsten. Additionally, the isotherms tend to be parallel to the anode surface because it is assumed to maintain a constant temperature of 1000 K. The other main limitation of the current model is that vaporization is neglected at the anode. Figure 11 shows the corresponding velocity vectors and the nature of the pool depression. As expected, the maximum velocity increases as the arc current increases (Table 3). It is interesting to note that the maximum velocity occurs near the tip of the cathode rather than near the anode surface. This is because the primary driving force for the cathode jet is the ) X B term. Figure 12 shows a typical vector plot of the current density, in this instance for the 200 A case. The strong divergence of current can be seen near the cathode tip which gives rise to the cathode jet. The relatively high conduc- 4 6 RADIAL LOCATION (MM) 2 tivity of the anode adjacent to the plasma causes the current density vectors to be nearly perpendicular to the deformed anode surface. Consistent with this observation, the anode surface is very nearly a line of constant electric potential. Figure 13 shows the surface shear stresses generated by the plasma gas as it passes radially outward over a flat anode surface. The maximum shear stress shown is for a 6.3-mm (0.25-in.) long 300-A arc which is seen to be nearly 100 N/m 2 . This shear stress is of the same order of magnitude as those caused by the surface tension gradients and may be a significant factor in determining the free surface shape. Previous investigators (with the exception of very recent work by Matsunawa, Ref. 35) have neglected the gas shear stress and this may be the reason previous models have been unable to predict the surface depressions based on arc pressure alone (Ref. 5). Modeling of Weld Pool Circulation with a Deformed Free Surface Figure 14 shows the computational domain employed in simulating the weld pool. The welding arc strikes the surface of the workpiece, adding heat to it via the radiative, convective and electronic contributions. The heat absorbed will raise the 4 6 RADIAL LOCATION (MM) ^^1 -i—i—i—i—I—i—i \~ 1—i—r - D 60 — 300 4.0 x » + - 40 — AMP ARC m m DEPRESSION Total Flux CJrad Qelectron 20 2 4 6 RADIAL LOCATION (MM) 2 Fig. 9 - Anode heat flux for the four cases studied. A - 200 A; B-260 A; C - 280 A; D- 300 A 352-s | SEPTEMBER 1990 4 6 RADIAL LOCATION (MM) temperature of the surface, and a molten pool develops. This pool will grow until the heat input equals the heat loss by radiation, convection, conduction and vaporization. Flow in the pool is driven by a combination of buoyancy, electromagnetic, surface tension and impinging plasma arc forces. The transport processes of importance are listed below: 1) Current and heat flux distributions due to the arc. 2) Interaction of the arc with surface (surface shape). The complete modeling effort requires a dynamic coupling between the arc and the weld pool. In this first attempt, the free surface has been assigned on the basis of the studies of Lin and Eagar (Ref. 5). The primary focus of the present effort is to investigate the type of flow field genera'ed by the bimodal heat source. Work is currently in progress to represent the two-way interaction between the weld pool and the welding arc. Basic assumptions for the model are: 1) Laminar flow is assumed since the 3) Convective heat transfer due to fluid flow in the molten weld pool. 4) Thermal conduction into the solid workpiece. 5) Convective and radiative heat losses from surface. 6) Heat and mass losses due to vaporization. In addition, t w o other processes that are of concern include (1) transient solidification and melting of the pool and (2) the free surface behavior as a result of surface ripples or surface deformation. 2 UJ s a. o—I UJ > UJ a o ce < UJ to Radial Position (mm) Radial Position (mm) 10. io. 2 3 4 5 6 7 8 - 2 3 4 5 6 7 8 9 12000 K 13000 K 14000 K 15000 K 17000 K 19000 K 21000 K - 12000 13000 14000 15000 17000 19000 21000 23000 Q. o > ui 1 - 11000 K 1 - 11000 K 2 Q K K K K K K K K X o oc < UJ co UJ cr 2 a O -i UJ > Ul a i oDC <t ui V) Ul 280 A 200 A ce, »•« i- ?. 0. 10. f 0. 4. • i - 1 - 11000 K > X TJ O • 6. 2 3 4 5 6 7 8 9 - 12000 13000 14000 15000 17000 19000 21000 23000 B K K K K K K K K > X ES' ' >- 4^ JWW\ / | \\\ TJ o o .3 4^ fi ./ttll\\\ l\Yv 'J 7 6. • i 1 2 3 4 - a. ' i 11000 12000 13000 14000 z HT • K K K K 1 UJ D 15000KK 65 - 17000 " 7 - 19000 K 8 - 21000 K 9 - 23000 K O —i UJ > UJ a — o < u V) u a ^ i- Z s 8. TJ. 2 a. T \ ^"^5" > ,) a o -i UJ > u a oia < 12. 260 A 14.^ f 300 A 12. Fig. 10 - Temperature contours for the four cases studied. A - 200 A; B - 260 A; C - 280 A; D - 300 A. WELDING RESEARCH SUPPLEMENT 1 353-s ui Ui a Radial Position (mm) Radial Position (mm) 3. > X > x T3 o 00 T3 o o 00 0 o 3 O 1\5 CS 3 u 280 A 2 1 ' n , t 4 1 fi 1 ' 1 a i ' p 1 D B > > ~ o 2 4 o 00 «:<^ ' 4.-77, / ' c ' fi s - • 1 j j | • > ( ' < • • • t , • < V • Hi? I I1' 1' . <' [ 1 5' V o p 1 '/ GO >—. vim < • • p. Wi J 3 CO ? ^ r r : "'.'.'. IP Hi 3 12 f 00 300 A 14 260 A 3 - if > ^r Fig. 11 - Velocity plots for the four cases studied. A - 2 0 0 A; B-260 A; C-280 A; D-300 A. size of the pool is small. 2) The surface is assumed to be a gray body. 3) All transport, physical and electrical properties of the liquid and solid are assumed constant, independent of temperature, with the exception of the surface tension, which is assumed to be a linear function of temperature. These assumptions may be readily relaxed without undue computational difficulty. 4) Vaporization is simulated, as a first approximation, by putting an upper boundary on liquid temperature, which is 500 K below the boiling point (Refs. 1,36). for the system depicted in Fig. 14 are: conservation of mass d(rur) dr J3u r For incompressible laminar flow, the relevant governing transport equations 354-s I SEPTEMBER 1990 dur dP • - 5r ' dr dUr <WJr ~7- + r)Z - K u 2 3uz[ 1 du, dP <92uz I (18) - Tr) + JrB„ - Kuz conservation of thermal energy fr3T (17) 2 u Prgl3(l 3u r l d-Ur r 3uz I <92uz (16) conservation of radial momentum ar Governing Transport Equations duz dz f<5u2 a \d2l 5T , 1dT + d2T [ |5? rlF 5?J_ AH Mi cn + 3T] (19) at r conservation of axial momentum The buoyancy forces are calculated via the Boussinesq approximation while the Lorentz forces are calculated in a manner identical to that described for the arc, except that the electrical conductivity is assumed constant for the workpiece. Note that in the weld pool modeling, the origin is taken to be the bottom of the workpiece, while in the welding arc modeling the origin is taken to be the electrode. Radial Position (mm) 200 A > X Special Source Terms and Boundary Conditions o The enthalpy method is used to simulate the melting phenomena. Here, a drag term is included in the momentum equation to inhibit flow when the temperature falls below the liquidus temperature. A latent heat term is added to the energy equation for phase change. In particular, the method proposed by Hirt (Ref. 37) is chosen for its simplicity. The temperature-dependent drag term is incorporated into the momentum equation via — Ku r and — Ku z where -v O 3 co CO + o 3 3 * # T > TL; k = 0 Fig. 12 —Current density vector plot for the 200-A case (1-mm depression). Ts < T < TL; k = k (JL - T) TL-TS (20) T < T s ; k = co which, by using the Darcy-type relationship, i.e., flow through porous media, represents flow phenomena in the mushy region. For convenience of comparison, the K term is defined as a function of drag via DRC DRG = 1 1.0 + A t • K than Fig. 15B. The latent heat term is added to the energy equation via (AHi/C p ) • (<3f|/3t) where f|_, the fraction of liquid, has been linearized for simplicity. It can be used to simulate true volume fraction liquid if the phase diagram of the alloy is known. Here, it is given as: = + ffX) (23a) df dr - • ( * ) - - ( » ) • T > TL; fL = 1 (S) Ts < T < TL; (21) Thus, DRG = 1.0 for pure liquid and 0.0 for solid state. The DRG term is also a function of the time step; for At = 10 _3 s, Kmax is chosen as 104. The effect of DRG as a function of T for t w o different values of Kmax is shown in Fig. 15. Figure 15A represents drag in the mushy zone better dur (T - T5) fL — (22) TL-TS T < Ts; fL = 0 The other t w o driving forces, surface tension and the plasma arc, are added as boundary conditions. For surface tension: (23b) where dy/dl has been defined as the surface tension coefficient. The arc force is the stress imposed upon the free surface to deform it. It may be calculated from the J X B term in the plasma jet or obtained from experimental measurements (Ref. 38). In this study, the effect of arc force is not included since w e have assumed a Arc Heat Flux 100 Vaporization V Convection and Radiation 3 2 J3 2 4 8 RADIAL LOCATION (UM) Fig. 13 - Shear stress distribution for a flat anode. Convection and Radiation Fig. 14 —Modes of heat transfer in the workpiece. WELDING RESEARCH SUPPLEMENT I 355-s 1.0 • 0.9 0.8 - Kmax = lo 3 0.7 0.6 o 13 CC a IT Q 0.5 0.4 0.3 - 0.2 0.1 B 0.0 1523 1723 TT 1623 To Temperature (K) Fig. 15 —Drag as a function of temperature between p r e d e f i n e d p o o l surface shape. W e h a v e also assumed that the f r e e surface shear is d u e only t o surface tension. T h e effect of m o m e n t u m transfer of the gas jet m a y b e i m p o r t a n t a n d an initial estimate has indic a t e d that d e p e n d i n g u p o n the arc curr e n t , the gas shear stress can b e in t h e same o r d e r of m a g n i t u d e as the surface tension shear stress. Thus, o n e may n o t readily ignore the gas jet as has b e e n d o n e in t h e past. • i 1523 Tc the liquidus and solidus temperature. This drag is used to model flow in the mushy zone. It should b e r e m a r k e d (Ref. 39) that f o r l o w current values ( < 2 0 0 A) the w e l d p o o l depression calculated f r o m purely m o m e n t u m considerations agrees w e l l w i t h measurements (Ref. 5). H o w e v e r , f o r higher current values, b o t h magnetic pressure a n d the arc pressure c a n n o t account f o r t h e experimentally o b s e r v e d d e e p depressions. Since the arc pressure, surface shear stress, shear stress d u e t o surface tension ( w h i c h seeks t o resist the d e - f o r m a t i o n ) , and the metallostatic head are all of the same o r d e r of m a g n i t u d e , o n e m a y speculate that a p r o p e r a c c o u n t i n g f o r the effect o f gas-induced surface shear may p r o v i d e the explanation for the d e e p depressions o b s e r v e d at high current levels. T h e input heat and current fluxes are o b t a i n e d f r o m the calculations described previously. T h e y serve as b o u n d a r y c o n ditions f o r t h e free surface. It should b e stressed that b y this p r o c e d u r e w e h a v e b e e n able t o p r o v i d e an i m p o r t a n t linkage b e t w e e n t h e arc a n d the w e l d p o o l b e havior. This is t h e first t i m e that this has b e e n d o n e explicitly. T h e c o n v e c t i v e heat loss at surfaces o t h e r than the w e l d p o o l is given by the standard e q u a t i o n o,out 200 A (a) 1723 1623 Temperature (K) <b) — hc(TSLJrf T a m b) (24) w h e r e h c is calculated f r o m an empirical correlation f o r natural c o n v e c t i o n . This is given as (Ref. 40): vertical w a hc= 1.42 hxrzis&y (25a) b o t t o m wall hc= ( 1.59 Ta A M Tsurf (25b) L In a d d i t i o n , the usual b o u n d a r y c o n d i tions are z e r o velocities at all solid surfaces, a n d t e m p e r a t u r e a n d velocity gradients are z e r o at the axis of s y m m e t r y . T h e physical p r o p e r t i e s of steel are listed in Table 4. Solution Technique (c) (d) Fig. 16 — Temperature profile in weld pool of the solidus line (1523 K) as a function of time for the four cases studied. A - 200 A;B-260A;C—280 A; D — 300 A. The axis of symmetry is on the righthand side with the origin at the bottom right-hand corner. The arc is at the top right and the nature of the blockage cells is indicated. The steps match those given in Fig. 8. The dimensions of the plots are given in Table 4. The numerals on the contour lines indicate the time in seconds. 3 5 6 - s I SEPTEMBER 1990 The g o v e r n i n g equations and b o u n d a r y conditions w e r e solved b y the c o m m e r cial fluid a n d heat f l o w numerical p a c k a g e called PHOENICS (Ref. 41). This uses the f i n i t e - v o l u m e a p p r o a c h . Four variable meshes are used t o simulate t h e f o u r arc currents previously d e s c r i b e d . T h e y are listed in Table 5. \N - ? \YV\VCvJ 280 A Fig. 17 — Velocity plots of the weld pool. The coordinate configuration is similar to that described in Fig. 15. A-t = 3.0 s (200 A): um3X=12 cm/s; B-t = 3.0 s (260 A): Umax" 12 cm/s; C-t = 3.0 s (280 A):umax = 11 cm/s; D-t = 7.0 s (300 A): umax = 11 cm/s; F — temperature plot for 300-A arc at t = 3.0s <1> 1523K;<2> 1700K;<3> 1900K; <4> 2 WOK. The dimensions for A, B and E are given in Table 5. Figures C and D are partial plots and their dimensions are 10.95 X 6.3 mm and 10.95 X 6.9 mm, respectively. 3 The time step used is 10" s. PHOENICS uses the fully implicit method and is thus inherently stable. Although a limit of 30 sweeps was set for each step, the residual has been defined at 10 - 8 , thus the number of sweeps needed to satisfy the residual may be less than 30. It has been noted the maximum residuals when the 30th sweep is reached is no larger than 10~3. Results and Discussions of the Weld Pool Modeling The calculation starts by postulating a given (deformed) solid shape; then the computed heat and current flux given in Figs. 8 and 9 are used to calculate the temperature and the velocity field as shown in Figs. 16 and 17. Here the solidus temperature (1523 K) is plotted as a function of time. In all cases, dy/dl is assumed to be + 10" 4 N/m-K. Because of the somewhat arbitrarily imposed initial conditions, these computed results are meaningful only for larger times (i.e., t > 3 s); indeed, only these results are given here. From Fig. 16 it seems clear that the hump is heated more quickly than the center of the pool because of the higher heat flux there. As a result, this region will reach peak temperature (2500 K) first and surface tension flow may be reduced. Previous models (Refs. 2, 42) have indicated that the highest velocities are encountered near the edge of the pool. The reduced flow can affect the geometry of the weld bead. This may have important Table 4—Physical Properties of the Workpiece implications depending on how long the anode is heated for a spot weld or how long a point is heated in a moving weld. Figure 16 also indicates the surface profile of the weld pool. Figure 17 shows the velocity plots of all four cases at t = 3.0 s except Fig. 17D which is shown at t = 7.0 s so as to depict the nature of the flow more clearly. A similar result is obtained at t = 3.0 s for the 300-A case, but the flow field is not so pronounced. Figure 17B shows a larger molten pool than that shown in 17A since a higher current is used, which will provide more melting. However, in contrast to 17D, Fig. 17C shows a deeper pool penetration at the center even though the Cp g hc Kmax k TL Ts = = = = = = 753 J/kg-K 9.81 m/s 2 80 W / m 2 - K 104 S- 1 15.48 W / m - K 1723 K 1523 K /3 7 AH L ( n p Tamb = = = = = = = 10" 4 K" 1 1.2 N/m 247 k|/kg 0.4 0.006 kg/m-s 7200 k g / m 3 300K surface is only depressed by 1.8 mm (0.07 in.). In addition, Fig. 17D has been heated for 7.0 s and has a surface depression of 4.0 mm (0.16 in). An important feature of these computed results is how the shape of the weld bead geometry affects the flow field. Since 5Y/<9T is positive, the flow is directed toward the direction of increasing surface temperature (normally toward the center of the pool). In Figs. 17A-C, the usual inward flow is obtained. The bimo- WELDING RESEARCH SUPPLEMENT | 357-s dal heat flux at the free surface causes a bimodal surface temperature distribution which can be seen in Fig. 17E. This will produce three loops; one directly below the hump, a second near the center of the pool, and a third in-between. This is seen in Fig. 17D. Notice that in Fig. 9D the maximum is t w o times the minimum heat flux. The consequence is the three velocity loops formed due to the strong surface tension-driven flows. Furthermore, in contrast to Figs. 17 A - C , here, the strong surface flows are able to bring the hotter fluid from the outside inward and a single loop is formed. It also implies that the hump is not heated fast enough (the maximumminimum heat flux ratio < 1.6 for the three cases) so to as to produce bimodal surface temperature profile. Instead, a unimodal surface temperature is produced and thus the three isothermal loops are not seen in Figs. 1 7 A - C The results in Fig. 17 indicate the importance of surface tension-driven flow in controlling pool circulation. Since these flows are driven by the temperature gradient, the ability to effectively describe the surface temperature is critical. In this model, the peak surface temperature has been taken empirically at 500 K below the boiling point of the workpiece, which is 2500 K (Refs. 1, 36). This implies that there are certain regions at the weld pool surface where 6T/5r equals zero and thus (mm) 16 r 14 12 10 fc: Cb) (d (d) te) Marangoni-driven flow is absent. This is one of the limitations of this model. Although this limitation can be overcome by assuming simple vaporization kinetics such as that due to Langmuir, this method tends to overpredict the rate of heat loss by a factor of ten. A more accurate vaporization model, which is currently under development, takes into account both Langmuir vaporization and mass transfer across the anode boundary layer. It is important to note that the latter model can only be developed provided the arc characteristics (plasma velocity and temperature) are known. This model will be incorporated in subsequent studies. While the specific nature of these computed results (i.e., the absolute values of the melt velocity, the number of recirculating loops and the like) will, of course, depend on the values assigned to the properties of the system, some very important general conclusions may be drawn from these computed results. These are the following: 1) For high current levels, surface tension and gas shear may play equally important roles in affecting melt circulation. 2) In the case of significantly deformed weld pools, quite complex circulation patterns may develop due to the nature of the (bimodal) heat flux. 3) The final, solidified shape of the weld bead may be also markedly affected by these phenomena. Modeling of Transient Weld Pool Collapse One of the intriguing free surface problems in welding is the collapse of the weld pool once the arc is extinguished. This is a somewhat simpler situation than the establishment of the transient free surface shape for continuous welding operations, ih) tg> Ij) Fig. 18 — Transient collapse of finger penetration 3 mm deep and 2 mm in diameter. The origin is at the bottom-left-hand corner, and the computational domain is 6 mm radial (12 grids) by 76 mm axial (32 grids). The liquid surface is at 10 mm. A—Oms, umax = 0 cm/s; B-5 ms, umax = 68 cm/s; C— 10 ms, umax = 32 cm/s; D— 15 ms, umax = 48 cm/s; E—20 ms, umax = 125 cm/s; F— 145 ms, Umax = 69 cm/s; C— 150 ms, um 56 cm/s; H— 160 ms, umax = 43 cm/s; I— 165 ms, umax = 35 cm/s; j - 180 ms, umax = 122 cm/s. Table 5 --Variable Mesh for the Cases Tested for the Weld Pool Modeling * Arc Current (A) Surface Depression (mm) Grids<a> (radial X axial) Domain Dimensions (mm) Real time simulated (s) CPU time (h) (MicroVax II) 200 260 280 300 1.0 1.2 1.8 4.0 17X22 17 X 22 17 X 35 17 X 30 10.95 X 6.2 10.95 X 6 . 2 10.95 X 12.3 10.95 X 12.5 3.0 3.0 3.0 7.0 42.6 44.4 52.9 150.0 (a) The number of grids is less than those given in Table 2 because PHOENICS automatically includes the nodes at the walls. 358-s | SEPTEMBER 1990 Fig. 19 — Schematic representation of liquid metal filling in the crater of a finger penetration weld after the arc is extinguished (Ref. 45). Fig. 20-Surface depression at 300 A of a stationary arc weld on Type 304 stainless steel. A — Top surface during welding showing deep narrow depression; B — cross-section of same weld. The porosity at the bottom of the finger penetration indicates that the depression penetrated to the very bottom of the weld (Ref. 5). nonetheless, it is an important intermediate step in reaching this goal. In order to illustrate the nature of this problem, let us illustrate the transient collapse of a keyhole. The material considered was steel, under isothermal conditions. This may seem an absurd oversimplification, but was thought to be a necessary first step in understanding the physical phenomenon of collapse. Governing Transport Equations and Solution Technique The governing transport equations include continuity (Equation 16) and momentum (Equations 17, 18). In addition, the volume of fluid equation (Ref. 43) employed to handle free surface is given by df 3t + 3F Ur 5r + SF Uz dz = ° ing assumptions made. The behavior shown in Fig. 18 is remarkable because it depicts the formation of a liquid droplet and the transient establishment of an occluded gas space. This behavior mimics very well the experimental observations of Lin and Eagar (Refs. 5,45), shown in Figs. 19 and 20. The purpose of modeling the collapse of the weld pool is to determine whether the pool surface will freeze before it flattens out. Such analysis may provide some insight into ripple formation and the uneven surface morphology frequently found in welding. Work is currently proceeding to represent the combined effect of heat transfer and weld pool collapse and such a sequence is shown in Fig. 21; these results are preliminary at this stage, but are indicative of the types of novel modeling efforts currently underway. Somewhat similar work, albeit with much smaller deformation and 0.2 mm (0.008 in.) pool radius dimension, recently has been published by Paul and DebRoy (Ref. 18). The way in which the deformation was handled was not discussed in detail. IRAN'SIENT fOOL BEHAVIOR WITH BUOYANCY AND P O S I T I V E ^ (26) where F = 1 fluid cell 0.0 < F < 1.0 surface cell 0.0 void cell (27) In addition to the boundary conditions given previously, there are additional boundary conditions needed to handle the free surface such as surface pressure due to curvature and the effect of wall adhesion. These are discussed in detail in Nichols, et al. (Ref. 43), and will not be reiterated here. t=0.0s ; u m a x = 5 8cm/s t=0.02s ; Umax='»4 cm/s t=0.08s; u m a x 5 ? c m / s t=0.5s, un-,ax=2 8 cm/s Fig. 21— Transient collapse of deep depression with solidification. The domain is 15 X 14 mm with a maximum axial and radial depression of 6.0 and 2.5 mm, respectively. dy/6T= 10~4 N/m-K. Results and Discussions of Weld Pool Collapse These calculations used the SOLA-VOF (Ref. 43) algorithm, and employing a coarse 12 X 32 grid ( 6 X 1 6 mm) required about t w o hours on a CYBER 205 supercomputer. Figure 18 shows the computed results indicating the very strong free surface disturbances and wave motion. Calculations not actually depicted here showed that a flat surface would be established after about 0.5 s. W e note that an order of magnitude analysis, using an expression cited by Batchelor (Ref. 44) would give a result within a factor of 3, which seems quite reasonable, in view of the simplify- WELDING RESEARCH SUPPLEMENT | 359-s Conclusions A n extensive set of calculations has b e e n p r e s e n t e d t o describe the b e h a v i o r o f the arc w e l d p o o l interface in arc w e l d i n g o p e r a t i o n s . T h e ultimate o b j e c tive of this w o r k is t o p r o v i d e a full r e p r e sentation of the t w o - w a y interaction b e t w e e n the w e l d i n g arc a n d t h e w e l d p o o l . T h e results r e p r e s e n t e d h e r e p r o v i d e a partial a t t a i n m e n t of this goal. Calculations are p r e s e n t e d t o describe the b e h a v i o r of a w e l d i n g arc w h i c h is m a d e t o i m p i n g e o n the t o p o f b o t h flat a n d d e f o r m e d w e l d p o o l surfaces. T h e c o m p u t e d results f o r a flat a n o d e agree w e l l w i t h measurements r e p o r t e d in t h e literature, b o t h regarding the heat flux a n d t h e current distribution. (The arc pressure is also w e l l represented). Calculations p e r f o r m e d f o r d e f o r m e d a n o d e surfaces h a v e s h o w n that the current and heat flux distributions m a y b e v e r y m a r k e d l y a f f e c t e d b y the shape o f t h e free surface of the w e l d p o o l . I n d e e d , it has b e e n s h o w n that t h e generally postulated Gaussian-type distribution may b e t r a n s f o r m e d t o a b i m o d a l distribution c u r v e . T h e v e r y i m p o r t a n t finding of this w o r k is that, particularly f o r high-current o p e r a t i o n s , t h e heat flux falling o n t h e w e l d p o o l surface m a y n o t be specified i n d e p e n d e n t l y o f the w e l d p o o l b e h a v ior — as has b e e n d o n e b y virtually all p r e vious investigators. Calculations p e r f o r m e d t o describe t h e shape a n d circulation of d e f o r m e d w e l d pools have relied o n the previously c o m p u t e d heat flux a n d current distribution as t h e b o u n d a r y c o n d i t i o n s ; thus p r o v i d i n g a first-order c o u p l i n g b e t w e e n the arc a n d w e l d p o o l calculations. H o w e v e r , the d e f o r m e d w e l d p o o l shape still had t o b e d e d u c e d f r o m previously r e p o r t e d experimental data. These calculations h a v e s h o w n that t h e melt circulation in these systems m a y b e m a r k e d l y a f f e c t e d b y b o t h the f r e e surface shape of the w e l d p o o l and also b y t h e nature of the incident heat flux. These findings suggest that t h e f l o w patterns in w e l d pools m a y b e e v e n m o r e c o m p l e x than previously t h o u g h t . Finally, calculations h a v e b e e n p r e sented describing t h e transient collapse o f a w e l d p o o l , o n c e the arc has b e e n extinguished. In these preliminary results, isot h e r m a l c o n d i t i o n s have b e e n p o s t u l a t e d as a first, rather a p p r o x i m a t e representat i o n ; nonetheless, e v e n these preliminary results w e r e able t o c a p t u r e s o m e essential features of experimentally o b s e r v e d w e l d p o o l collapse, including t h e e n t r a p m e n t of gases a n d the expulsion of liquid droplets. It is t h o u g h t that t w o i m p o r t a n t issues have b e e n raised b y the w o r k d e s c r i b e d here. O n e , is that f o r t h e first t i m e it has b e e n possible t o bring a b o u t a c o u p l i n g b e t w e e n t h e b e h a v i o r o f the w e l d i n g arc a n d the w e l d p o o l , such that t h e heat flux 360-S | SEPTEMBER 1990 falling o n t h e p o o l is directly calculated f r o m t h e arc properties a n d that the arc b e h a v i o r is a f f e c t e d b y t h e shape o f the f r e e surface of the w e l d p o o l . The s e c o n d , rather m o r e i m p o r t a n t point, is that this interaction m a y i n v o l v e quite substantial effects. M o r e specifically, w h e n the w e l d p o o l is d e f o r m e d , t h e arc characteristics m a y b e m a r k e d l y m o d i f i e d ; i n d e e d , the previously e m p l o y e d p r o c e d u r e o f specifying t h e heat flux falling o n the p o o l i n d e p e n d e n t l y o f t h e p o o l characteristics, is p r o v e n t o b e questionable f o r m a n y situations. By the same t o k e n , f r e e surface d e f o r m a t i o n m a y m a r k e d l y m o d i f y b o t h the shape of the melt-solid interface a n d the circulation w i t h i n t h e weld pool. It should b e stressed that these c o m p l e x p r o b l e m s are far f r o m being fully s o l v e d . A great deal m o r e w o r k will b e r e q u i r e d t o p r o v i d e a faithful representation o f the d y n a m i c c o u p l i n g b e t w e e n the w e l d i n g arc a n d the w e l d p o o l surface, particularly t h e f r e e surface d e f o r m a t i o n and the f r e e surface instabilities. Acknowledgments T h e a u t h o r s w i s h t o thank the U.S. D e p a r t m e n t o f Energy, Basic Energy Sciences Division, f o r financial s u p p o r t of this investigation u n d e r Grant N o . DE-FG0287ER45289, a n d in part t o Florida State University S u p e r c o m p u t e r C o m p u t a t i o n s Research Institute, w h i c h is partially f u n d e d by the U.S. D e p a r t m e n t of Energy t h r o u g h C o n t r a c t N o . DE-FC-85ER25000. References 1. Oreper, G. M., Eagar, T. W., and Szekely, ]. 1983. Convection in arc weld pools. Welding yoivma/62(11):307-s to 312-s. 2. Kou, S„ and Wang, Y. H. 1986. Weld pool convection and its effect. We/ding Journal 65(3):63-s. 3. fin, M. L, and Eagar, T. W . 1983. Transport Phenomena in Materials Processing, ASME-PED, 10, p. 63. 4. Heiple, C. R., and Rober, ). R. 1982. Mechanism for minor element effect on GTA fusion zone geometry. Welding Journal 61 (4):97-s to 102-s. 5. Lin, M. L, and Eagar, T. W . 1985. Influence of arc pressure on weld pool geometry. Welding lournal 64(6): 163-s to 169-s. 6. Oreper, G. M., and Szekely, ]. 1984. / Fluid Mech., 147, p. 53. 7. Chan, C. L., Mazumder, )., and Chen, M. M. 1984. Met. Trans. A., Vol. 15A, p. 2175. 8. Kou, S„ and Sun, D. K. 1985. Met. Trans. A., Vol. 16A, p. 203. 9. Correa, S. M., and Sundell, R. E. 1986. Modeling and Control of Casting and Welding Processes, S. Kou and R. Mehrabian, eds., TMS-AIME, Warrendale, Pa., p. 211. 10. Paul, A., and DebRoy, T. 1986. Advances in Welding Science and Technology, S. A. David, ed., ASM International, Materials Park, Ohio, p. 29. 11. Oreper, G. M., Szekely, I., and Eagar, T. W . 1986. Met. Trans. 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Appendix B e F g fl h he he K Kma: k kb L Nu w P 2 Magnetic flux density (Wb/m ) Azimuthal magnetic field (Wb/m 2 ) Heat capacity ()/kg) Electronic charge (C) Void fraction (-) Gravitational acceleration (m/s 2 ) Fraction of fluid (—) Plasma enthalpy (J/kg) Heat transfer coefficient (W/m 2 -K) Plasma enthalpy at the edge of the boundary layer (J/kg) Plasma enthalpy at the anode temperature (J/kg) Current (A) Current density (A/m 2 ) Anode current density (A/m 2 ) Cathode current density (A/m 2 ) Radial current density (A/m 2 ) Axial current density (A/m 2 ) Drag index in source term (kg/m 3 s) Maximum drag index (kg/m 3 s) Thermal conductivity (W/m-K) Boltzmann constant (J/K) Characteristic length (m) Nusselt number at the anode (—) Pressure (Pa) Prw Qa Qc Qe Qioniz Qr Q r ij qout Rc Re w r t\ j Sr SR T Tamb Te Teiec Tf TL Tr Ts Tsurf Tw t ur ure U2 V arc Vc ve Plasma Prandtl number at the anode temperature (—) The anode heat flux (W/m 2 ) Convection contribution to the anode heat flux (W/m 2 ) Electron contribution of the anode heat flux (W/m 2 ) Heat source from the cathode fall to ionize the plasma (W/m 2 ) The radiative contribution to the anode heat flux (W/m 2 ) Heat flux received at surface S, from volume Vj located the distance r r j away (W/m 2 ) heat loss from anode (W/m 2 ) Cathode spot radius (m) Reynolds number at the anode (—) Radial coordinate (m) The direction vector from Sj to Vj (m) The differential surface in the radiation view factor relation (J/m2s) Radiation source (W/m 3 ) Temperature (K) Ambient temperature (K) Plasma temperature at the edge of the boundary layer (K) Electron temperature (K) Film temperature (K) Liquidus temperature (K) Reference temperature for Boussinesq approximation (1523 K) Solidus temperature (K) Surface temperature (K) Wall temperature (K) Time (s) Radial component of velocity (m/s) radial velocity at the edge of the boundary layer (m/s) Axial component of velocity (m/s) Arc voltage calculated (includes approximate cathode fall voltage) (V) Cathode fall voltage (V) Plasma velocity parallel to the anode surface at the edge of the boundary layer (m/s) The differential volume in the radiation view factor relation (m3) Work function of the anode material (V) Axial coordinate (m) Q. o > UJ Q X Greek Symbols a fi y AH At A fi Mw Me Mo P Pe Pr Pw a Tzx 8 *c 4> 4>6 * dy Thermal diffusivity (m 2 /s) Coefficient of thermal expansion (K- 1 ) Surface Tension (N/m) Latent heat of fusion (J/kg) Time step (s) Del operator Viscosity (kg/m-s) Viscosity at the anode temperature (kg/m-s) Viscosity at the edge of the boundary layer (kg/m-s) Magnetic permeability of free space (H/m) Density (kg/m 3 ) Density at the edge of the boundary layer (kg/m 3 ) Reference density for Boussinesq approximation (7200 kg/m 3 ) Density at the anode temperature (kg/m 3 ) Electric conductivity (ohm-m) n Shear stress (Pa) Azimuthal direction (radian) Work function of the cathode material (V) Electric potential (V) Thermal diffusion coefficient for electrons (A/m-K) The angle ry makes with the normal to surface > (radian) Surface tension coefficient (N/m-K) 5T o 0C < UJ tn m OC z UJ £ Q. o > UJ a x o DC < Ui tn UJ ac z UJ S Q. o -I UJ > X o oc < UJ tn UJ ce Ul 5 Q. O _J UJ > UJ WRC Bulletin 349 December 1989 o O OC < Ui tn This bulletin contains two reports that evaluate the PWHT cracking susceptibility of several Cr-Mo steels and several HSLA pressure vessel and structural steels. Ui rc -•>, r- Z UJ Postweld Heat Treatment Cracking in Chromium-Molybdenum Steels £ oa. -I By C. D. Lundin, J. A. Henning, R. Menon and J. A. Todd UJ > Postweld Heat Treatment Cracking in High-Strength Low-Alloy Steels UJ a By R. Menon, C. D. Lundin and Z. Chen Publication of this report was sponsored by the University Research C o m m i t t e e of the Welding Research Council. The price of WRC Bulletin 349 is $35.00 per copy, plus $5.00 for U.S., or $10.00 for overseas, postage and handling. Orders should be sent with payment to the Welding Research Council, 345 E. 4 7 t h St., Room 1 3 0 1 , New York, NY 10017. WELDING RESEARCH SUPPLEMENT 1361-s o oc < UJ tn